[FOM] "Proof" of the consistency of PA published by Oxford UP

Rob Arthan rda at lemma-one.com
Thu Mar 5 16:31:57 EST 2015

On 4 Mar 2015, at 20:33, Richard Heck <richard_heck at brown.edu> wrote:

> So we're being given an informal semantic consistency proof. But exactly what the cash value of the proof is, it seems to me, isn't obvious until we know exactly what sorts of assumptions it is employing. Somewhere, obviously, there are some strong assumptions being deployed. The fact that the presentation obscures where they are is not a virtue.

I couldn’t agree more about the lack of virtue in the presentation.
The following points do seem to be clear:

1) McCall wants to give a semantics couched in some kind of naive physics,
in which certain properties of arrangements of congruent cubes
(referred to as “blocks”)  in physical space are taken to be self-evident.

2) The semantics for the quantifiers assumes that an additional block
can be added to an arrangement of blocks without unwanted intersections.
So the naive physics assumes that the universe has infinite volume and
contains an unbounded amount of matter.

3) The proof of the induction axiom via what McCall calls the method
of “finite descent” assumes that a process that removes blocks from
an arrangement one at a time will always terminate with an empty
arrangement. So the naive physics disallows the creation of infinite
arrangements despite the abundance of space and matter implied
by the semantics for quantifiers.

I, personally, find it fairly irritating that the article claims to be doing
something very new, when the naive physics adds nothing to the
naive arithmetic intuition that PA is obviously consistent, that one
gets, for example, by thinking about concrete decimal representations of
numbers and the arithmetic operations on them. 



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